A similar question and partially complete proof have been posted here: How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?, and in fact my question comes from this post.
Let $V$ be a normed space and $W\subset V$ be a linear subspace. I have proved the following:
Lemma 1: The followings are equivalent:
- $V$ is a Banach space;
- For every sequence $\{x_{k}\}\subset V$ such that $\sum_{k=1}^{\infty}\|x_{k}\|<\infty$ the series $\sum_{k=1}^{\infty}x_{k}$ is convergent, i.e. there exists $y\in V$ such that $$y=\lim_{M\rightarrow\infty}S_{M}\ \ \ \ S_{M}:=\sum_{k=1}^{M}x_{k}.$$
Basically, it shows that $V$ is a Banach space $\iff$ absolutely convergence implying convergence. And following the post above, I want to use this lemma, to prove the following statement:
If $V$ is a Banach space and $W$ is closed, then $(V/W, \|\ \cdot\ \|_{V/W})$ is a Banach space, where $$\|[x]\|_{V/W}:=\inf_{y\in W}\|x-y\|_{V},\ \ \text{and}\ \ [x]\ \ \text{is the equivalence class}.$$
I have proved the most of it:
Let $([x_{n}])_{n=1}^{\infty}$ be a sequence in $V/W$ such that $\sum_{n=1}^{\infty}\|[x_{n}]\|_{V/W}<\infty$. We want to show that $\sum_{n=1}^{\infty}[x_{n}]$ converges to some $[x]\in V/W$, and then Lemma 1 concludes the proof.
Recall the definition of the quotient norm, for each $n\in\mathbb{Z}_{\geq 1}$, we have $\|[x_{n}]\|_{V/W}=\inf\{\|x_{n}-y\|_{V}, y\in W\}.$ It then follows from the definition of infimum that, for each $n\in\mathbb{Z}_{\geq 1}$, there exists $y_{n}\in W$ so that $$\|x_{n}-y_{n}\|_{V}\leq \|[x_{n}]\|_{V/W}+\dfrac{1}{2^{n}}.$$
It then follows that $\sum_{n=1}^{\infty}\|x_{n}-y_{n}\|_{V}<\infty.$
But $(x_{n}-y_{n})$ is a sequence in $V$ and $V$ is a Banach space, by Lemma 1, we know that the series $\sum_{n=1}^{\infty}x_{n}-y_{n}$ converges to some $x\in V$.
But then how could I conclude that $\sum_{n=1}^{\infty}[x_{n}]$ converges to $[x]$ in $V/W$? I tried to evaluate $$\Bigg\|\sum_{n=1}^{\infty}[x_{n}]-[x]\Bigg\|_{V/W}=\inf_{y\in W}\Bigg\|\sum_{n=1}^{\infty}(x_{n}-x)-y\Bigg\|_{V}$$ But I don't know how to use the above result.
Thank you for any help!!
Best Answer
You are almost done:
Recall that $|| z+W ||_{V/W} \leq ||z|| $ for all $z \in V$ (i prefer to use the notation $z+W$ for the coset), thus
$$||\Big (\sum_{k=1}^n (x_k-y_k) -x \Big) +W ||_{V/W} \leq || \sum_{k=1}^n (x_k-y_k)-x||_V \xrightarrow{n \to \infty} 0.$$ Notice that the LHS is equal to $$ || \Big(\sum_{k=1}^n x_k -x +W \Big) - \Big (\sum_{k=1}^n y_k +W \Big )||_{V/W} = || \sum_{k=1}^n x_k - x +W ||_{V/W}, \ \text { since } \ \sum_{k=1}^n y_k \in W .$$ So, $ \sum_{k=1}^n x_k +W \xrightarrow{n\to \infty} x$+W.